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I have seen small stones being thrown into water in such a way that they repeatedly bounce off the water surface and after certain bounces , the stone sinks into water .

QUESTION: Why do the stones bounce off the water surface instead of sinking into water ? Surely the water surface isn't rigid enough to provide enough reaction force, is it?

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It is called stone skipping:

An early explanation of the physics of stone-skipping was provided by Lazzaro Spallanzani in the 18th century.

The stone generates lift in the same manner as a flying disc, by pushing water down as it moves across the water at an angle. Surface tension has very little to do with it. The stone's rotation acts to stabilize it against the torque of lift being applied to the back.

Research undertaken by a team led by French physicist Lydéric Bocquet discovered that an angle of about 20° between the stone and the water's surface is optimal. Bocquet and his colleagues were surprised to discover that changes in speed and rotation did not change this fact. Work by Hewitt, Balmforth and McElwaine has shown that if the horizontal speed can be maintained skipping can continue indefinitely.

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The rock has an angle of attack, so the water at the bottom of the stone changes from static to downward flow, so the pressure of the water increases. So the water has a lift on the stone. The lift makes the stone move upward. After moving for a period of time, the stone leaves the water. After moving in the air for a period of time, under the action of gravity, the stone falls back to the water. Because the stone has an angle of attack, so the bottom of the stone has a lift, so after a period of time the stone left the water again...

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